Saving and Investment

Classical Model

Learning Objectives

• Learn how to derive and shift the investment demand curve

• Learn how to derive and shift the saving supply curve.

• Understand the role of interest rates in the classical model.

• Understand the impact of technology on investment demand.

• Understand the impact of “animal spirits” on investment demand.

• Understand the impact of social changes on the supply of saving.

Business Cycles and Investment: Classical Model

• Classical economists argue that business fluctuations are caused by a series of shocks to technology that alter the productivity of labor in a random way from one year to the next.

• These shocks are transmitted to the capital market through changes in investment, causing saving, investment and the interest rate to fluctuate during the business cycle in an apparently random way.

• Business cycles are a necessary and unavoidable feature of market economies.

Investment Volatility: Classical Model

• Investment spending is volatile.– Investment fluctuates much more than GDP over

time.

• In classical theory, output and employment are determined by fundamentals such as technology, which can also influence investment.– For example, new technology can cause an

investment boom in new machines that are designed to exploit the invention.

Investment Volatility: Keynesian Model

• In Keynesian theory, changes in investment represent changes in the mass psychology of investors or “animal sprits.”– Greenspan’s use of “irrational exuberance” is a

recent example of this Keynesian idea.

• According to Keynesians, business cycles can be avoided if investment is more efficiently coordinated.

• Keynesians favor government policies to stabilize the business cycle.

Consumption

• Unlike investment, consumption fluctuates less than GDP over time.

• Consumption is smooth because households borrow and lend in the capital market in an effort to redistribute their income more evenly over time.

• Keynesian economists agree that consumption is smooth, but they argue that it could be even smoother.

“Smooth” Consumption• Keynesian economists argue that consumption is

not as smooth as it could be because households with low incomes do not have easy access to capital markets.

• If people who prefer to consume more than their income are credit constrained, their optimal decision is to consume all their income.

• Therefore, the presence of credit constrained individuals implies that aggregate consumption will fluctuate more than otherwise since the credit constrained individuals’ consumption will fluctuate equally with GDP.

Theory of Investment

• Assumptions:– Individuals consume what they produce.– As time passes, individuals allocate their

produced commodities between consumption goods and investment goods.

Intertemporal Production Possibilities Frontier

Investment Consumption0

Future Income

Current IncomeAB

D

E

The maximum attainable resourcesavailable for consumption andinvestment = 0A.

If an individual invests 0B andconsumes BA, he will have 0Eavailable to divide betweenconsumption and investment inthe future.

The Production Function

• The intertemporal production function slopes up because the more an individual invests today, the greater his income in the future.

• The intertemporal production function’s slope rises at a decreasing rate because of the law of diminishing returns.– For every increase in capital, output increases by

smaller amounts.

Investment and Profit Maximization

• Firms invest up to the point where the output produced by an extra unit of investment is equal to its cost.– Firms equate the marginal product of

capital investment with the real rate of interest.

• At this point, they are investing in the profit maximizing amount of capital.

Profit: Definition

• A firm’s profit is the value of its produced output minus the accrued principal and interest on loans needed to purchase current investment goods.

• Profit = Value of Future Sales – Cost of Borrowing

= Y – (1+r)I– Y = value of output tomorrow

– r = market rate of interest

– I = investment of resources today

Investment and Production FunctionYt+1

0

0

(1+r)

I

I

I

A

B

Panel A shows the intertemporal productionfunction: the amount of output that can beproduced in the future (Yt+1) for any giveninvestment today (I).

Panel B shows the relationship betweenthe real interest factor (1 + r) and the quantityof capital investment demanded (I).

Investment and Production FunctionYt+1

0

0

(1+r)

I

I

I

I1

(1+r)1

Y

I1

A

B

Panel A: As the firm buys additional capital,the extra output produced by the last unit ofcapital decreases.

The additional output is the marginal productof capital investment = slope of the productionfunction.

The firm invests up to the point where (1+r)just equals the marginal product of capital investment. At this point, profits are maximized.

Panel B shows the investment schedule. At higher rates of interest, investment is less and at lower rates of interest, investment is greater.

(slope =(1+r))

Deriving the Investment Schedule: Math

• Max = AI – (½)(I)2 – (1 + r)IAI – (½)(I)2 = Total Product in t+1(1 + r)I = Total Borrowing Cost A = Technology induced shifts in

investment.Find the derivative of profit with respect to I, set it equal to zero, and solve for the marginal product of investment.

• d/dI = A – I – (1 + r) = 0• A – I = (1 + r)

Households and Saving

• Intertemporal utility theory forms the basis for most modern explanations of how income is divided between consumption and saving.

• This theory argues that, given the choice, families would prefer that consumption be evenly distributed over time.

Intertemporal Budget Constraint

• Assumptions:– Households consume part of its income and

saves part.– They put their savings into the capital market

by lending to another household or firm and receive future resources with interest.

• The amount of additional goods that households can buy in the future grows with the rate of interest.

Present Value

• The capital market can be used to transfer resources from the present to the future.

• It can also be used to transfer resources from the future to the present.– When an individual borrows against future

income, he/she borrows its present value.

Present Value

• Present value tells us how much an expected future payment is worth today.– For example, if we expect to inherit $10,000

next year, but wish to spend it today, we can borrow some amount less than $10,000 today.

– The amount that we can borrow is determined by the interest rate.

Present Value Formula

• The formula for present value can be found by rearranging the compounding formula.

FV = PV(1 + i) Compounding

• Solve for PVFV/(1 + i) = PV Present Value

Intertemporal Budget Constraint

• Households can use capital markets to redistribute resources over time.

• The intertemporal budget constraint places a bound on the amount of consumption that is available over a household’s lifetime.

• C1 + C2/(1 + r) <= Y1 + Y2/(1 + r)

Intertemporal Budget Constraint

C1 + C2/(1 + r) <= Y1 + Y2/(1 + r)

C1 = Present consumption

C2/(1 + r) = Present value of future consumption

Y1 = Current resources

Y2/(1 + r) = Present value of future resources.

Intertemporal Budget Constraint

• C1 + C2/(1 + r) <= Y1 + Y2/(1 + r)– The left hand side of the constraint represents

the value of the goods that a person consumes at every point in life, valued in terms of current consumption goods.

– The right hand side of the constraint represents the value of the income that a person earns at every point in life, valued in terms of current consumption goods.

Deriving Supply of Saving Curve: Math

• Household Utility: Utility = Utility of C1 + Utility of C2

U = – (2 – C1)2/2 + C2

• Two-Part Household Budget Constraint: 1) Future Consumption = Interest + Principal

C2 = (1 + r)S 2) Saving = Income – Current Consumption

S = Y – C1

Deriving Saving Supply: Math

• Incorporate the budget constraints into the utility function. S = Y – C1 implies – C1 = – Y + S

C2 = (1+r)S

• Household UtilityUtility = Utility of C1 + Utility of C2

U = – (2 – Y + S)2/2 + (1 + r)S

Deriving Saving Supply: Math

• Maximize Household Utility• Max U = – (2 – Y + S)2/2 + (1 + r)S

• Present utility given S = – (2 – Y + S)2/2

• Future utility given S = (1 + r)S

• Find the derivative of U with respect to S, set it equal to zero, and solve for S.

Deriving Saving Supply: Math

• Max U = – (2 – Y + S)2/2 + (1 + r)S

• du/dS = – (2 – Y + S) + (1 + r)

Utility lost by saving one more unit = – (2 – Y + S)

Utility gained by saving one more unit = (1 +r)

• S = Y – 1 + r

The Saving Supply Curver

S0

S

As interest rates rise, individualssubstitute future consumption forpresent consumption.

Saving and Interest Rates

• Substitution Effect– When the real rate of interest increases, current

consumption becomes relatively more expensive, causing households to want to substitute consumption today for consumption tomorrow. Saving increases.

• Wealth Effect– When the real rate of interest increases, households

may save more or less. Higher rates of interest cause wealth grow faster, but higher interest rates also reduce the present value of accumulated wealth.

Capital Market Equilibrium

r

0

S

I

The upward sloping line representssaving.

The downward sloping line representsinvestment.

At the intersection of S and I, the model is in equilibrium

S,I

Equilibrium: Math

• Previous Results:– Investment Demand = A – I = (1 + r)– Saving = Y – 1 + rSet I = S and solve for equilibrium r.

I = A – (1 +r) = A – 1 – rS = Y – 1 + rA – 1 – r = Y – 1 + rA – 1 – Y + 1 = 2rr = (A –Y /2)

Equilibrium: Math

• Previous Results:r = (A –Y /2)I = A – 1 – r

Substitute equilibrium r in I or S to solve for equilibrium I or S.

I = A – 1 – (A – Y/2)I = A – 1 – A/2 + Y/2I = 2A/A – A/2 + Y/2 – 1I = A/2 + Y/2 – 1 I = (A + Y)/2 - 1

Capital Market Equilibrium

r

0

S

I

S,I

At r1, investors want to borrow more fundsthan individuals are willing to save. Theexcess demand pushes interest rates up.

At r3, investors want to borrow fewer funds than savers are willing to save. The excess supply demand pushes interest rates down.

At r2, investors want to borrow the sameamount that individuals are willing to save.

r3

r2

r1 A B

C D

Say’s Law and the Classical Theory of Interest

• Output in the classical model is determined via the production function and depends on what happens in the labor market.– Involuntary unemployment is impossible.

• What if there is a glut of goods that cannot be purchased by a fully employed labor force?

Say’s Law and the Classical Theory of Interest

• Say’s Law asserts that supply creates its own demand or every act of production constitutes a demand for something.– There is no problem if we are in a barter

economy, but once money is introduced a loophole emerges if people decide to save some of their income rather than spending it all.

Saving and Investment

Firms Households

Income

Expenditures

Financial Markets SavingBorrowing

Investment

Say’s Law and the Classical Theory of Interest

• By making saving and investment a function of the interest rate, the classical theory of saving and investment preserves the integrity of Say’s Law.– In classical theory, the rate of interest unites

decisions to save by households with decisions to invest by business firms.

Say’s Law and the Classical Theory of Interest

• If individuals and households try to save more out of current income than firms want to spend on investment at the prevailing rate of interest, interest rates fall.– As rates fall, savings fall, consumption rises,

and investment rises.

Say’s Law and the Classical Theory of Interest

• If individuals and households try to save less out of current income than firms want to spend on investment at the prevailing rate of interest, interest rates rise.– As rates rise, savings rise, consumption falls,

and investment falls.

Business Cycles and Investment

• Classical Theory– Changes in fundamentals cause business cycles.

• For example, a new invention makes investment more productive, leading to an increase in capital investment.

• Keynesian Theory– Irrational swings of optimism and pessimism

are more important driving forces in the stock market than fundamentals.

Productivity and InvestmentY

Y2

Y1

0

0

I

Ir

Y1

Y2

I1(r1)

I1(r1)

I2(r1)

r1

I1 I2

S

I2(r1)

A

B

An increase in productivity shifts investment demand to the right. Why?

We begin at point 1. The interest rate is r1,and the marginal product of investmentequals the slope of a tangent line at Point 1. Investment equals I1(r1).

An increase in productivity shifts up theproduction function from Y1 to Y2.

If interest rates remain at r1, investment increases to I2(r1). We move to point 2.

At point 2, I>S, causing r to rise to r2.

Equilibrium occurs at Point 3.

1

1

(1+r1)

(1+r1)

2

2

r2

3

Saving and Investment: Open Economy

• In an open economy, domestic saving does not have to equal domestic investment.– It can be less than domestic investment or more

than domestic investment.• When I < S, a country may lend abroad.

• When I > S, a country must borrow from abroad.

Saving and Investment: Open Economy

• Y = CNAT + INAT + NX

• Y – CNAT – INAT = NX

• SNAT – INAT = NX– If SNAT = INAT, NX =O, trade balance– If SNAT > INAT, NX >0, trade surplus– If SNAT < INAT, NX <0, trade deficit

Looking at X - M

• X – M then is income minus consumption vis a vis the rest of the world.– If X > M, a country has excess funds to lend to

the ROW, or S > I.– If X < M, the country’s trading partner has

excess funds to lend to it or domestically S < I.

Saving and Investment: Open Economy

• Definitions:– USA demand for capital from the rest of the world

equals the difference between national domestic investment and national domestic saving for different values of the world interest rate.

– The world supply of capital equals the amount that other countries are willing to lend in world capital markets at different values of the world interest rate.

• The world supply of capital curve slopes up because countries are willing to lend more as the interest rate rises.

USA Demand for Capital from the Rest of the World Curve

r

r1

0

0 S,I

K

r1DK

USA

SKROW

A

NB1

S1 I1

NB1>0

A

B

At r1 in Panel A, national domestic investment is greater than national domestic saving.

The difference in domestic investment over domestic saving is paid for by borrowing NB1 in the world capital market.

At r1 in Panel B, net borrowing from the ROW equals NB1 = I1 – S1.

Point A represents one point on the USAdemand for ROW capital curve.

I

S

rw

USA Demand for Capital from the Rest of the World Curve

r

r1

r2

0

0 S,I

K

r1

r2

DKUSA

SKROW

A

NB1

S1 I1

NB2

NB1>0

A

B

At r2 in Panel A, national domestic investment is less than national domestic saving.

The difference in domestic saving overdomestic investment results in negativeborrowing = NB2 in the world capital market.

At r2 in Panel B, negative net borrowing from the ROW equals NB2 = I2 – S2.

Point B represents another point on the USA demand for ROW capital curve.

I

S

B

NB2<0

S2I2

Equilibrium in the World Capital Market

r

r1

r2

0

0 S,I

K

r1

r2

DKUSA

SKROW

A

NBE NB1

S1 I1

NB2

NB1>0

A

B

I

S

B

NB2<0

S2I2

req

C

Equilibrium occurs at the point where USAdemand for capital from the world just equals the world supply of capital to the USA.

Point C in Panel B represents equilibrium in the world capital markets.

At this point, USA investment exceeds domestic savings. The difference is made up by borrowing NBE from abroad

NBE>0